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With constant coefficients, $\lambda =(\mu -r)/\sigma$ is constant, so the stochastic exponential $Z_t=e^{rt}{Y}_t$ is a true martingale. Under its equivalent martingale measure $\mathbb{Q}$, the stock follows the Black-Scholes model $d{S}_t=S_t(rdt+\sigma d{W}_t^{\mathbb{Q}})$. The Gaussian exponential moment gives the value of the power optionThe option delta is ${\partial }_{s}V=V/(2s)$, so the replicating strategy holdsHalf the wealth value is in the stock and half in the bank account, with continuous rebalancing. To check the self-financing portfolio property under the original measure, the Itô formula gives $d{V}_t=\frac{1}{2}(r+\mu )V_{t}dt+\frac{1}{2}\sigma V_{t}d{W}_t={\theta }_{t}d{S}_t+{\eta }_{t}d{B}_t$. Also $V_T=\sqrt{S_T}$ and $V_t>0$.

The payoff is unbounded, so the bounded-payoff restriction of part (c) is not invoked automatically; the Black-Scholes model has the necessary finite Gaussian moments, and the explicit strategy attains $V_0=\mathbb{E}[Y_T\sqrt{S_T}]$. The supermartingale bound from part (b) therefore proves its minimality. The cost is independent of the physical drift $\mu$.

A contingent claim whose payoff depends on the underlying price history, rather than only its terminal price. Asian options use averages, barrier options use whether a threshold has been crossed, and lookback options use running extrema. In the Black-Scholes model, the Brownian Martingale representation theorem still gives a replicating strategy; its portfolio wealth need not be a function of the current stock price alone. Adding a running integral or extremum can give a finite-dimensional state for pricing.

For $s\ge 0$ and $\epsilon >0$, integrate over $0\le K\le s$ to obtain the displayed identity. Tonelli theorem then expresses the moment of a nonnegative random variable as the same integral of expected call payoffs, even when the moment is infinite. This is a static representation across strikes rather than a dynamic replicating strategy in a restricted finite-asset market.